Insights into $(k,ρ)$-shortcutting algorithms

TL;DR

This work studies transforming arbitrary graphs into $(k,\rho)$-graphs by adding shortcuts, formalized as the $(k,\rho)$-Minimum-Shortcut Problem ($k\rho$-MSP). It proves $k\rho$-MSP is $\mathcal{NP}$-hard for all fixed $k\ge 3$ and $\rho=\Theta(n^{\varepsilon})$, and derives lower bounds on the approximation ratios of existing heuristics. To address practical instances, it introduces preprocessing steps and a family of heuristics ($k\rho$-DP-PC, $k\rho$-DP-SA, and MinHash) plus an exact ILP formulation that can solve instances optimally. The authors benchmark these methods on diverse random graph models and real-world networks, showing substantial empirical gains in graphs with powerlaw degree distributions while also highlighting challenges in social-network-like graphs; they conclude with open questions on the parameter regimes and the possibility of sublinear approximation guarantees. Overall, the paper links a fundamental graph-augmentation problem to NP-hardness and provides both theoretically-grounded bounds and practical algorithms that improve shortest-path preprocessing in parallel settings. $k$, $\rho$, and their interplay emerge as central knobs for designing scalable, structure-exploiting PSSSP methods.

Abstract

A graph is called a $(k,ρ)$-graph iff every node can reach $ρ$ of its nearest neighbors in at most k hops. This property proved useful in the analysis and design of parallel shortest-path algorithms. Any graph can be transformed into a $(k,ρ)$-graph by adding shortcuts. Formally, the $(k,ρ)$-Minimum-Shortcut problem asks to find an appropriate shortcut set of minimal cardinality. We show that the $(k,ρ)$-Minimum-Shortcut problem is NP-complete in the practical regime of $k \ge 3$ and $ρ= Θ(n^ε)$ for $ε> 0$. With a related construction, we bound the approximation factor of known $(k,ρ)$-Minimum-Shortcut problem heuristics from below and propose algorithmic countermeasures improving the approximation quality. Further, we describe an integer linear problem (ILP) solving the $(k,ρ)$-Minimum-Shortcut problem optimally. Finally, we compare the practical performance and quality of all algorithms in an empirical campaign.

Figures (8)

• Figure 1: Transformation of edge $\left\{u,v\right\}$ for the input $G=(V,E)$. Each edge in $E$ implies their own nodes in $V_\text{edges}\xspace$ and $V_\text{subs}$ (left); each original node in $V$ has its own pitchfork-gadget (right). By construction $w_{u,v}$ is too far from the leaves of either pitchfork. By adding, e.g., the canonical shortcut (see \ref{['def:trivial']}) between $u$ and $b_u$, the leaves become available for the $(k, \rho)$-ball of $w_{u,v}$.
• Figure 2: Schematics for the central components of $k\rho$-DP-PC (a), (b) and $k\rho$-DP-SA (c).
• Figure 3: A graph for which the optimality of the solution of $k\rho$-MSP for $k=2$, $\rho=6$ and a single node $v$ already depends on the encoding of a specific shortest path tree with fewest hops. Thus until it is clear which tree preserves the optimality of the solution an optimal algorithm has to encode all of them.
• Figure 4: The comparison between the optimal algorithm and the heuristics for the $k\rho$-MSP problem for $k=2,\ \rho=n-1$ on several random graph classes for varying $n$. $\sigma_h$ and $\sigma_b$ describe the average approximation factor for $k\rho$-DP-PC-SA+MH and $k\rho$-DP respectively, while $\sigma_{max} = \max_i\sigma_i$ is the maximum approximation factor wittnessed on up to 50 sampled instances.
• Figure 5: Miscellaneous results on including average running time, average factor of edge increases and solavbility on Gilbert random graphs.

Theorems & Definitions (25)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Theorem 6
• Lemma 7
• Definition 8
• Lemma 9
• Corollary 10